3.2.5 The Fourier Transform

The Fourier transform of the complex muon polarization $\tilde{P}(t)$ gives a good approximation of the actual internal field distribution. The Fourier transform is

$$n(B) = \int_{0}^{\infty} \, \tilde{P} (t) e^{ -i (\gamma_{\mu} B t + \theta )} dt \, .$$ (29)

Due to the finite counting rates, the Fourier transform contains statistical noise. The noisy or distorted portions of the asymmetry spectrum can be eliminated from the Fourier transform through ``apodization''. Apodization is achieved by multiplying the asymmetry spectrum by a weighting function which varies between one and zero [48]. For example, the Fourier transform can be apodized with a Gaussian function so that  

$$n_A (B) = \int_{0}^{\infty} \, \tilde{P} (t) e^{ -i (\gamma_{\mu} B t + \theta )} e^{-\sigma_A^2 t^2/2} dt \, .$$ (30)

Unfortunately, this apodization procedure also broadens the Fourier transform. The apodization parameter $\sigma_A$ is chosen to provide a compromise between the statistical noise in the spectrum and the additional broadening of the spectrum which such a procedure introduces. The Fourier transform will also appear broader than the actual field distribution in the sample because the muon spin precession signal is measured over a finite time interval. Because of the finite number of recorded events, the integral in Eq. (3.34) is replaced with a sum. Examples of the Fourier transform of the muon precession signal in the vortex state will be presented in the next chapter. Since $\omega_{\mu} \! = \! 2 \pi \nu_{\mu} \! = \! \gamma_{\mu} B$, the Fourier transform can be presented as a function of either magnetic field B or the muon precession frequency $\nu_{\mu}$.